To the downvoter, I'm aware that I'm not an active member of the site and I may not be following the explicit and implicit rules. As such, I would greatly appreciate it if you could tell me how this question could be improved (or doesn't belong at all)!
– HookedJul 25 '13 at 15:37

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I'm sorry about you receiving a downvote. It happens to the best of us. But well done on your attitude and approach!
– Mari-Lou AJul 25 '13 at 15:42

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I'm most comfortable using #4 (except "need", not "needs" since that refers to "they") - but 2 and 3 are OK too.
– Kristina LopezJul 25 '13 at 15:45

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@Mari-LouA I'm a big fan of the SE sites in general, and I realize that each site has it's own culture. The downvote is less important than knowing where I silently erred!
– HookedJul 25 '13 at 15:46

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Why wouldn't you just change the "he" to "she"? (Sure, I realize Alice Cooper might object, but that seems like a good way to resolve this particular issue.) But, to answer your general question, I would reword this, and use: I know the OP thinks all the eigenvalues are needed... Voilà! No plural pronoun, no gender-specific pronoun.
– J.R.Jul 25 '13 at 20:47

For what seems to be the core of your specific question - I think #4 sounds better than #3; it's not 'more correct' gramatically, just a matter of style.

I'll repeat here what has been beaten to death elsewhere (including the question you linked): "He" is gender neutral. "She" is not; I'd think it a greater mistake to use "she", thus declaring an assumption about the gender of "alice" (esp. on the internet). Because "he" is gender-neutral, it doesn't actually suggest such an assumption.

"He/she" is a little off, as others have said "he or she" is a little clunky (although correct).

There are others (like the answer you linked), who know more than I, who will insist that singular "they" is OK. I'm sure those folks have a good argument, but to my old-fashioned, prescriptivist ears, it's not.

I thought of the same example as @Mari-LouA - "the captain". "The captain thinks he needs full engine power." / "The captain thinks they needs full engine power." Doesn't the former sound better?

In regards to the statement of the captain, I feel that "the captain ... he" is expressively a gendered statement, at least to my ears (so a personal assessment). I'll freely admit to "he" being gender-neutral in most cases, in fact that's how I used it in the question that started all of this! Since however I was corrected, I wanted to understand the implications of each choice and how to avoid a mistake like this in the future.
– HookedJul 26 '13 at 13:59

An honest question, when you say "prescriptivist ears", are you implying that in a formal sense, the singular "they" is not correct?
– HookedJul 26 '13 at 14:02

@Hooked But does "the captain ... they" sound right? // I think it would be a valid response to say to 'Mr. Protip', "'He' is gender-neutral, and I am not making an assumption about alice's gender". Note that when I say "valid", I mean you would have a sound grammatical justification; that doesn't mean this is the only correct way, or that this is the optimal way to commnicate with people (bickering about grammar when not on ELU).
– hunter2Jul 28 '13 at 5:59

What I mean by that is: The way I was taught, "they" ("singular they") is not correct, and "he" is a corect choice. That said, the link you posted provides a cogent counterpoint. If you check @nohat's profile, you'll see his position. I think it's fair to say that he would disagree with the very idea of 'correctness' in this context, and that he is well-qualified to make that case. As a counterpoint to that, there are other professionals/scholars who would take more of what nohat calls a 'prescriptivist' view; I suspect JohnLawler might 'lean more prescriptivist'.
– hunter2Jul 28 '13 at 6:01

(Now that I just followed the link to the context: I think your response was fine, other than referring to "OP" as a pronoun, as MariLou noted.)
– hunter2Jul 28 '13 at 6:06